I think rain is pretty cool so that probably wouldn't be a problem. If you're watering the soil/mulch and not the foliage I'd think it would be totally fine. If you're sprinkling seedlings in the spring it may not be as good since they need as much heat as they can get.
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I've seen where people run channels along the surface of the soil to slowly heat the water before it runs into their garden. Saw that in Peru in the Andes. It would be easy enough to lay black plastic down and pump the water over a shallow swale lined with the plastic, letting it come up to temperature before it flows into your garden.
Or not . . . too much work?
Easier would be a big 500 gallon tank painted black: pump the well water up into an elevated tank, wait a day, and let it gravity feed onto your garden through a simple garden hose. If you watered in the evening, the water will have had the full day to get warmer (perhaps as much as 10 degrees or more), and the ground will not be so warm at that time, thus not shocking the roots with the cold water.
Water has tremendous thermal mass. Once warm (after sitting in a solar heated tank all day), it will hold that heat into the evening. It might actually extend the soil warmth for a hour or more, thus turning a problem into a solution -- a longer growing day.
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I myself would not be overly concerned with irrigation water temperature or any sort of thermal shock. If it were me, I'd use it right out of the ground. The biggest concern is keeping the soil moisture consistent and not letting things dry out before irrigating.
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Actually it did affect the overall production in my case. Cold water slows down growth by shocking plants. I have heavy/ clay soil and so water remains in the first 1-2 of inches cooling it down significantly. I doubt well drained soils will be affected as much as larger mass of soil will partake in heat exchange. I used to water straight from the well, now I direct some to a black water tank at the top of the property (where it warms up a bit), then I use it in the garden.
Alternatively you can see that constant cold-water supply of yours as a resource. It is a very sought after resource surprisingly. You can:
-direct cold water to your house initially, where it will cool the house in summer months (the problem is humidity can condense on cold-pipes, basic air conditioning knowledge will suffice, it is the most simple AC )
-build a pond, have koi or trout or similar cold-water fish
-build a pond near your pergola and plant large canopytrees around to create shade. It will be the most pleasant place in hot summer months.
-in very hot but humid climates you can use cold water to catch water. Lay pipes on the surface around the garden (stupid cheap plastic pipes), let cold water circulate by a very weak pump. You would be surprised!
-most importantly! grow lettuce or other cold hardy plants in summer by spraying cold water couple of times. As an example:
Large rocks or beds of rock mulch, have a high heat capacity. I often direct hose water over rocks. This prevents erosion and buffers temperature. It also reduces evaporation, since the water doesn't absorb into surface mulch, where it can quickly be lost. The wetter spots are capped with an effective sun shield.
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